Introduction to Random Matrices

Giacomo Livan, Marcel Novaes, and Pierpaolo Vivo have written a small but lively book on random matrices. The book assumes a moderate amount of mathematical sophistication, but does not assume any prior knowledge of random matrices. It covers a lot of ground in only 124 pages and gives frequent references for those wishing to pursue a particular topic further.

The book feels conversational. It has an informal tone, a sense of humor, and a pleasant mixture of exposition and detail. The pace is brisk, appropriate for a book aiming to be an introductory survey rather than a formal textbook. And unlike surveys that only survey results in a particular area, this book also goes into enough detail to give a taste of some of the common techniques in its area.

It should be no surprise that a random matrix is matrix-valued random variable, though the field would not be very interesting if one left it at that. The subject is concern with how random matrices behave as matrices, primarily their spectral properties, rather than simply as boxes of random variables. The book is primarily concerned with the distribution of eigenvalues, and to a lesser extent that of eigenvectors.

More specifically, the authors are primarily interested in Gaussian ensembles. The simplest case is that of Gaussian Orthogonal Ensembles, or GOEs. These are symmetric real matrices formed by averaging a matrix \(A\) and its transpose \(A^T\) where the entries of \(A\) are independent draws from a Gaussian distribution. Gaussian Unitary ensembles (GUEs) are Hermitian matrices constructed analogously from complex entries, and Gaussian Symplectic ensembles (GSEs) are the quaternion-valued analog.

The book intersperses theory and numerical examples. The MATLAB source code for the numerical examples is available online.

Introduction to Random Matrices mentions applications to physics, and goes into some applications in depth. A reader with a sufficient background in physics would get more out of these parts of the book, but the discussions are accessible to someone without the benefit of such a background.

The authors say in the preface that they intend to offer a “truly accessible introductory account of [random matrix theory] for physicists and mathematicians at the beginning of their research career” and they deliver.

John D. Cook is an independent consultant in applied mathematics.